Real Stable Polynomials and their Zeros

A polynomial in several variables is real stable if it has real coefficients and never vanishes when every variable is taken from the open upper half of the complex plane. In a talk for a reading group, I go through these polynomials and the operations that preserve stability, building toward Gurvits's short proof of the van der Waerden conjecture: the permanent of every n×n doubly stochastic matrix is at least n!/nⁿ. The proof tracks a single quantity (the capacity) through a sequence of partial derivatives, each of which keeps the polynomial stable and loses only a controlled factor.